In the same paper, Fuglede proved this conjecture when the tiling complement or the spectrum is a lattice in \(\mathbb {R}^n\) . 30 years later, Tao [1]} disproved this conjecture by constructing a non-tile spectral set in \(\mathbb {R}^5\) . Currently, the conjecture does not hold in both directions for \(\mathbb {R}^n\) , \(n\ge 3\)  [2]}, [3]}, [4]}, [5]}. However, this conjecture remains open in \(\mathbb {R}\)  and \(\mathbb {R}^2\) .
